Why does the Sun always rise in the East? Why does the Sun always rise in the East?
Earth is rotating on its own axis & also revolves around the Sun, then how come the Sun always rises in the East?
 A: It takes a year for Earth to revolve around the Sun, and only one day to rotate about itself. That is why you can, for most practical purposes, forget about the revolution (which causes the different seasons) and concentrate only on the rotation, at least fo sun raising purposes.
A: Whether the sun "rises" in the east depends on your position on earth, and the time of the year. In northern latitudes, during the summer, the sun rises significantly North of East, and in the winter it rises in the South. For example, today's sunrise/sunset directions in Umeå Sweden, look like this (source: www.suncalc.net)

The yellow line shows the direction of sunrise, the orange line the current direction of the sun, and the red line the direction at sunset.
As you can see, the sun never gets close to being in the East...
The reason for this is the fact that the earth's axis of rotation is not perpendicular to its plane of rotation about the sun (the ecliptic). The same mechanism that causes summer and winter in the higher latitudes gives rise to this changing direction.
But as for the fundamental question: the rotation of the earth about its axis is much faster than the rotation of the earth about the sun - so the rotation of the earth is dominating the direction of the sunrise. Now if the earth stopped rotating altogether, the sunrise would be in the "Westerly direction", since the direction of both rotations (seen from say the North Star) is in the counterclockwise direction. It would just rise and set only once a year... But the year is 365.24 days, and while that is so, the sun will rise "mostly in the east".
A small addendum: because the earth's orbit is elliptical, its angular speed relative to the sun changes a little bit with the seasons. This is enough to make a sundial "off" by up to 15 minutes, depending on the time of year. This is captured in the "equation of time" and shown, for example, in this graph (from http://upload.wikimedia.org/wikipedia/commons/0/02/Tijdvereffening-equation_of_time-en.jpg):

And just for your amusement - on October 25th, the sun briefly rises in the South on Svalbard (Spitsbergen) before disappearing for the winter...
A: At first ask yourself the question what do you mean by "East"? 
Other answers have already said that why you should only concentrate on earth spin rather than its orbital motion towards that sun.
See North, East, West and South are not absolute directions and changes with the latitude and longitude. So wherever you are on the earth surface, depending on your local position you have a fixed North, East, West and South directions with respect to your own co-ordinate system. But these directions which are fixed with respect to you  are changing with respect to an observer at the sun because of the earth's spin. Now as the earth is rotating counter-clockwise(as seen from the top of the North pole) you will always encounter rising sun in the direction East irrespective of which month it is. Try to visualize and you will get it.
So if in the planets Venus and Uranus we do define "North, East, West and South" in the same way as in the picture in the link, there sun rises in the West due to their Retrograde motion.
A: The question is quite geocentric, as it is dependent on the observer's location.


*

*On Venus, which has a retrograde axis of rotation, the Sun appears
to rise in the West.

*In Arctic regions, the Sun may not rise or set for up to six months
of the year, and even then, it may not be clear in exactly which
direction the sunrise/sunset occurs.

*Other suns in the sky (ie stars) rise and set at all points around
the compass.
A: The Sun does not rise, it is the horizon that goes down.
You say that Sun rises in the East (with a certain degree of oscillations due to the tilt of the axis) just because the Earth spins from West to East. The revolution affects the difference between sidereal time and solar time, and makes the solar day $\approx 4$ minutes longer
If the Earth spinned in the opposite direction the Sun would appear to rise from the West.

It does not address the question in any meaningful way. First of all,
  the answer ignores the problem “how west–east direction is defined”? - Incnis mrsi

Why should my post answer that off-topic and silly question? Incnis, you ask a new  question about that and I will address it.
Any definition is a convention, you can debate for hours which is the best for Venus. OP didn't ask about that, he knows the conventional definition and we all know what 'East' means. 
Some posts introduced foreign elements or useless nitpick in order to differentiate themselves, I just gave the basic answer.
A: It looks like the Sun rises in east and set in west, but actually it is altogether vice versa. Our earth rotates around the sun in west to east fashion. Since we see ourselves as stationary in our frame of reference(i.e. earth), the sun seems to rise and set in east to west fashion. Similarly we have winter and summer as seasons due to the revolution of earth around the sun.
A: 
Earth is rotating on its own axis & also revolves around the Sun, then how come the Sun always rises in the East?

Earth orbits around the Sun, but the motion can be expressed equivalently in Earth reference frame as Sun orbiting Earth. The orbital speed is not perfectly uniform, but close enough not to be distinguishable in normal life.
Earth also rotates. The plane of rotation is tilted (about 23.5°) compared to orbital plane, but let's ignore that for a moment. The rotation speed is even closer to uniform than orbital speed.
Orbit around rotating body can be expressed in the rotating reference frame, that is the reference frame of you as observer on Earth, as simple orbit with angular speed equal to sum of the two angular speeds. Since both angular speeds are almost uniform, so is the resulting orbit in your reference frame. Therefore you see the Sun move across the sky in always the same direction and always the same angular speed.
Now we established the Sun raises always at the same side (in reference frame fixed to place on Earth) we can call that side East and the opposite side West.
The angle between the two rotation axis cause the lateral shift moving the Sun higher above the horizon in summer (which is when your hemisphere is tilted towards the Sun) and lower in winter (which is when your hemisphere is tilted away from the Sun). The yearly variation make sundials be somewhat ahead or behind clock time depending on time of year (up to about 20 minutes).
A: 
(image credit: Gdr@Wikimedia)
Essential answer, assuming the observer near Earth’s equator, is: 


*

*When the Sun rises, the observer is near the centre of the leading Earth hemisphere (upper side of the blue disk in the image), and east points towards the Sun.

*At noon, the Sun is above the observer, and east points backward, against the orbital motion.

*At sunset, east points away of the Sun, because the planet made a half-revolution with respect to Sun and the observer is now near the centre of the trailing Earth hemisphere (lower side of the blue disk).

*At midnight, the Sun is under the observer, and east points forward, along orbital motion.


Year-round change of direction towards Sun in inertial frame (i.e. with respect to far stars) is not very important, assuming a low axial tilt (Earth has about 23°).
This is an overly simplified picture ignoring both observers at high latitudes (such as myself) and axial tilt. For a retrograde rotation leading and trailing hemisphere will swap, but even this doesn’t cause the central body to rise from different sides depending on time of year. Such dependence will happen only in the case of planetary rotation in a very different plane than the orbital plane, such as on Uranus.
